To solve the given problem, we will proceed with the steps necessary to find and simplify the difference quotient, and then take the limit as [tex]\( h \to 0 \)[/tex] to find the derivative:
a. Simplify the difference quotient [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex]
The function [tex]\( f(x) \)[/tex] is given by:
[tex]\[ f(x) = \frac{1}{-3x - 7} \][/tex]
First, we need to find [tex]\( f(x+h) \)[/tex]:
[tex]\[ f(x+h) = \frac{1}{-3(x+h) - 7} \][/tex]
[tex]\[ f(x+h) = \frac{1}{-3x - 3h - 7} \][/tex]
Next, we write the difference quotient:
[tex]\[ \frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{-3x - 3h - 7} - \frac{1}{-3x - 7}}{h} \][/tex]
To simplify this expression, we need to subtract the two fractions in the numerator:
[tex]\[ \frac{\frac{1}{-3x - 3h - 7} - \frac{1}{-3x - 7}}{h} = \frac{\frac{(-3x - 7) - (-3x - 3h - 7)}{(-3x - 3h - 7)(-3x - 7)}}{h} \][/tex]
[tex]\[ = \frac{\frac{-3x - 7 + 3x + 3h + 7}{(-3x - 3h - 7)(-3x - 7)}}{h} \][/tex]
[tex]\[ = \frac{\frac{3h}{(-3x - 3h - 7)(-3x - 7)}}{h} \][/tex]
Now, we simplify the fraction by canceling [tex]\( h \)[/tex]:
[tex]\[ = \frac{3h}{h(-3x - 3h - 7)(-3x - 7)} = \frac{3}{(-3x - 3h - 7)(-3x - 7)} \][/tex]
Thus, the simplified difference quotient is:
[tex]\[ \frac{3}{(3x + 7)(3h + 3x + 7)} \][/tex]
So,
[tex]\[ \frac{f(x+h) - f(x)}{h} = \frac{3}{(3x + 7)(3h + 3x + 7)} \][/tex]
b. Find the derivative [tex]\( f'(x) \)[/tex]
The derivative [tex]\( f'(x) \)[/tex] is the limit of the difference quotient as [tex]\( h \to 0 \)[/tex]:
[tex]\[ f'(x) = \lim_{h \to 0} \frac{3}{(3x + 7)(3h + 3x + 7)} \][/tex]
As [tex]\( h \to 0 \)[/tex], the term [tex]\( 3h \)[/tex] in the denominator becomes negligible:
[tex]\[
f'(x) = \frac{3}{(3x + 7)(3(x) + 7)} = \frac{3}{(3x + 7)^2}
\][/tex]
So, the derivative is:
[tex]\[ f'(x) = \frac{3}{(3x + 7)^2} \][/tex]
In summary:
a. The simplified difference quotient is:
[tex]\[ \frac{f(x+h) - f(x)}{h} = \frac{3}{(3x + 7)(3h + 3x + 7)} \][/tex]
b. The derivative of the function is:
[tex]\[ f'(x) = \frac{3}{(3x + 7)^2} \][/tex]
